LMFDB - Embedded newform 75.22.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,22,Mod(1,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 22, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	75.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$209.608008215$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 5 x^{9} - 15486550 x^{8} - 930225020 x^{7} + 80619202368510 x^{6} + \cdots + 47\!\cdots\!18 $ x^10 - 5x^9 - 15486550x^8 - 930225020x^7 + 80619202368510x^6 + 14206919536731394x^5 - 159833300365521427920x^4 - 53427455410418539963580x^3 + 93639331506281672292832825x^2 + 44269185428005803484908321595*x + 4729332062567824907768533377518
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{14}\cdot 5^{24}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2641.03$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2706.03 q^{2} +59049.0 q^{3} +5.22544e6 q^{4} -1.59788e8 q^{6} +9.90426e7 q^{7} -8.46523e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q-2706.03 q^{2} +59049.0 q^{3} +5.22544e6 q^{4} -1.59788e8 q^{6} +9.90426e7 q^{7} -8.46523e9 q^{8} +3.48678e9 q^{9} +6.27235e10 q^{11} +3.08557e11 q^{12} -7.95684e11 q^{13} -2.68012e11 q^{14} +1.19486e13 q^{16} -1.22879e13 q^{17} -9.43534e12 q^{18} -1.18746e13 q^{19} +5.84836e12 q^{21} -1.69731e14 q^{22} +3.16934e14 q^{23} -4.99864e14 q^{24} +2.15314e15 q^{26} +2.05891e14 q^{27} +5.17541e14 q^{28} -9.42196e14 q^{29} +4.96245e15 q^{31} -1.45804e16 q^{32} +3.70376e15 q^{33} +3.32515e16 q^{34} +1.82200e16 q^{36} +4.95881e16 q^{37} +3.21329e16 q^{38} -4.69843e16 q^{39} +3.21140e16 q^{41} -1.58258e16 q^{42} +4.99847e15 q^{43} +3.27758e17 q^{44} -8.57633e17 q^{46} -1.66127e17 q^{47} +7.05554e17 q^{48} -5.48736e17 q^{49} -7.25591e17 q^{51} -4.15780e18 q^{52} -5.56895e17 q^{53} -5.57147e17 q^{54} -8.38418e17 q^{56} -7.01180e17 q^{57} +2.54961e18 q^{58} +2.92119e18 q^{59} +3.85749e18 q^{61} -1.34285e19 q^{62} +3.45340e17 q^{63} +1.43970e19 q^{64} -1.00225e19 q^{66} -2.68566e19 q^{67} -6.42099e19 q^{68} +1.87146e19 q^{69} +3.71100e19 q^{71} -2.95164e19 q^{72} -4.95703e19 q^{73} -1.34187e20 q^{74} -6.20497e19 q^{76} +6.21229e18 q^{77} +1.27141e20 q^{78} +6.84806e19 q^{79} +1.21577e19 q^{81} -8.69014e19 q^{82} +1.44041e20 q^{83} +3.05603e19 q^{84} -1.35260e19 q^{86} -5.56358e19 q^{87} -5.30969e20 q^{88} -2.66839e20 q^{89} -7.88065e19 q^{91} +1.65612e21 q^{92} +2.93028e20 q^{93} +4.49544e20 q^{94} -8.60960e20 q^{96} -7.10430e20 q^{97} +1.48490e21 q^{98} +2.18703e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 645 q^{2} + 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} - 115357290 q^{7} - 314142735 q^{8} + 34867844010 q^{9}+O(q^{10}) $ 10 * q - 645 * q^2 + 590490 * q^3 + 10043205 * q^4 - 38086605 * q^6 - 115357290 * q^7 - 314142735 * q^8 + 34867844010 * q^9	$ 10 q - 645 q^{2} + 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} - 115357290 q^{7} - 314142735 q^{8} + 34867844010 q^{9} + 5976221790 q^{11} + 593041212045 q^{12} - 842570747430 q^{13} + 1605059874570 q^{14} + 6061861547825 q^{16} - 12910340404230 q^{17} - 2248975938645 q^{18} - 76155422176280 q^{19} - 6811732617210 q^{21} - 461780887241010 q^{22} - 82640920915920 q^{23} - 18549814359015 q^{24} + 286555331159670 q^{26} + 20\!\cdots\!90 q^{27}+ \cdots + 20\!\cdots\!90 q^{99}+O(q^{100}) $ 10 * q - 645 * q^2 + 590490 * q^3 + 10043205 * q^4 - 38086605 * q^6 - 115357290 * q^7 - 314142735 * q^8 + 34867844010 * q^9 + 5976221790 * q^11 + 593041212045 * q^12 - 842570747430 * q^13 + 1605059874570 * q^14 + 6061861547825 * q^16 - 12910340404230 * q^17 - 2248975938645 * q^18 - 76155422176280 * q^19 - 6811732617210 * q^21 - 461780887241010 * q^22 - 82640920915920 * q^23 - 18549814359015 * q^24 + 286555331159670 * q^26 + 2058911320946490 * q^27 - 1520428218531930 * q^28 - 2553653160731010 * q^29 - 5579671364724940 * q^31 - 29255856594217815 * q^32 + 352889920477710 * q^33 + 36285968426431010 * q^34 + 35018490530045205 * q^36 - 1797908431222890 * q^37 - 10684960744876320 * q^38 - 49752960064994070 * q^39 - 157319729506021440 * q^41 + 94777180533483930 * q^42 + 148924853700743400 * q^43 + 49118820773767290 * q^44 - 238453286038259540 * q^46 + 43171138083158100 * q^47 + 357946862537518425 * q^48 + 453692886452454950 * q^49 - 762342690529377270 * q^51 - 5077473324432504030 * q^52 - 1835165786666252970 * q^53 - 132799780201048605 * q^54 - 5146515624013530450 * q^56 - 4496901524087157720 * q^57 - 3684042125074285440 * q^58 + 4936135121536323630 * q^59 - 4443080871007408300 * q^61 - 39366851097921260280 * q^62 - 402225999313633290 * q^63 - 14877283608188093455 * q^64 - 27267699610694399490 * q^66 - 33664559574358232580 * q^67 - 44791395195934111290 * q^68 - 4879863739164160080 * q^69 - 74662724971386630180 * q^71 - 1095347988085476735 * q^72 - 16062862692682477620 * q^73 - 130249023546610518030 * q^74 - 258736483662380220720 * q^76 - 56216747719240740180 * q^77 + 16920805749647353830 * q^78 - 49130708010799643300 * q^79 + 121576654590569288010 * q^81 - 117185080045656261930 * q^82 - 118408383608453695560 * q^83 - 89779765876091934570 * q^84 - 1007271272794707382260 * q^86 - 150790665488005409490 * q^87 - 41803519849182967710 * q^88 - 572103976713991581780 * q^89 - 1669810835257196575140 * q^91 + 1206545757146297440260 * q^92 - 329474014415642982060 * q^93 - 1671319720124696594920 * q^94 - 1727529076031967757935 * q^96 + 1299275526640742173920 * q^97 + 2683959884079093505455 * q^98 + 20837796914288297790 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2706.03	−1.86860	−0.934302	−	0.356482i	$-0.883976\pi$
$2$	−2706.03	−1.86860	−0.934302	+	0.356482i	$0.883976\pi$
$3$	59049.0	0.577350
$3$	59049.0	0.577350
$4$	5.22544e6	2.49168
$5$	0	0
$5$	0	0
$6$	−1.59788e8	−1.07884
$7$	9.90426e7	0.132523	0.0662617	−	0.997802i	$-0.478893\pi$
$7$	9.90426e7	0.132523	0.0662617	+	0.997802i	$0.478893\pi$
$8$	−8.46523e9	−2.78737
$9$	3.48678e9	0.333333
$10$	0	0
$11$	6.27235e10	0.729133	0.364567	−	0.931177i	$-0.381217\pi$
$11$	6.27235e10	0.729133	0.364567	+	0.931177i	$0.381217\pi$
$12$	3.08557e11	1.43857
$13$	−7.95684e11	−1.60079	−0.800396	−	0.599471i	$-0.795378\pi$
$13$	−7.95684e11	−1.60079	−0.800396	+	0.599471i	$0.795378\pi$
$14$	−2.68012e11	−0.247634
$15$	0	0
$16$	1.19486e13	2.71680
$17$	−1.22879e13	−1.47831	−0.739155	−	0.673535i	$-0.764775\pi$
$17$	−1.22879e13	−1.47831	−0.739155	+	0.673535i	$0.764775\pi$
$18$	−9.43534e12	−0.622868
$19$	−1.18746e13	−0.444329	−0.222164	−	0.975009i	$-0.571312\pi$
$19$	−1.18746e13	−0.444329	−0.222164	+	0.975009i	$0.571312\pi$
$20$	0	0
$21$	5.84836e12	0.0765124
$22$	−1.69731e14	−1.36246
$23$	3.16934e14	1.59524	0.797621	−	0.603158i	$-0.206091\pi$
$23$	3.16934e14	1.59524	0.797621	+	0.603158i	$0.206091\pi$
$24$	−4.99864e14	−1.60929
$25$	0	0
$26$	2.15314e15	2.99125
$27$	2.05891e14	0.192450
$28$	5.17541e14	0.330206
$29$	−9.42196e14	−0.415875	−0.207937	−	0.978142i	$-0.566675\pi$
$29$	−9.42196e14	−0.415875	−0.207937	+	0.978142i	$0.566675\pi$
$30$	0	0
$31$	4.96245e15	1.08742	0.543711	−	0.839272i	$-0.317019\pi$
$31$	4.96245e15	1.08742	0.543711	+	0.839272i	$0.317019\pi$
$32$	−1.45804e16	−2.28926
$33$	3.70376e15	0.420965
$34$	3.32515e16	2.76238
$35$	0	0
$36$	1.82200e16	0.830561
$37$	4.95881e16	1.69535	0.847676	−	0.530515i	$-0.178001\pi$
$37$	4.95881e16	1.69535	0.847676	+	0.530515i	$0.178001\pi$
$38$	3.21329e16	0.830275
$39$	−4.69843e16	−0.924218
$40$	0	0
$41$	3.21140e16	0.373649	0.186825	−	0.982393i	$-0.440180\pi$
$41$	3.21140e16	0.373649	0.186825	+	0.982393i	$0.440180\pi$
$42$	−1.58258e16	−0.142971
$43$	4.99847e15	0.0352711	0.0176355	−	0.999844i	$-0.494386\pi$
$43$	4.99847e15	0.0352711	0.0176355	+	0.999844i	$0.494386\pi$
$44$	3.27758e17	1.81677
$45$	0	0
$46$	−8.57633e17	−2.98088
$47$	−1.66127e17	−0.460693	−0.230347	−	0.973109i	$-0.573986\pi$
$47$	−1.66127e17	−0.460693	−0.230347	+	0.973109i	$0.573986\pi$
$48$	7.05554e17	1.56855
$49$	−5.48736e17	−0.982438
$50$	0	0
$51$	−7.25591e17	−0.853503
$52$	−4.15780e18	−3.98867
$53$	−5.56895e17	−0.437398	−0.218699	−	0.975792i	$-0.570181\pi$
$53$	−5.56895e17	−0.437398	−0.218699	+	0.975792i	$0.570181\pi$
$54$	−5.57147e17	−0.359613
$55$	0	0
$56$	−8.38418e17	−0.369391
$57$	−7.01180e17	−0.256533
$58$	2.54961e18	0.777105
$59$	2.92119e18	0.744070	0.372035	−	0.928219i	$-0.378660\pi$
$59$	2.92119e18	0.744070	0.372035	+	0.928219i	$0.378660\pi$
$60$	0	0
$61$	3.85749e18	0.692376	0.346188	−	0.938165i	$-0.387476\pi$
$61$	3.85749e18	0.692376	0.346188	+	0.938165i	$0.387476\pi$
$62$	−1.34285e19	−2.03196
$63$	3.45340e17	0.0441745
$64$	1.43970e19	1.56093
$65$	0	0
$66$	−1.00225e19	−0.786617
$67$	−2.68566e19	−1.79997	−0.899987	−	0.435917i	$-0.856424\pi$
$67$	−2.68566e19	−1.79997	−0.899987	+	0.435917i	$0.856424\pi$
$68$	−6.42099e19	−3.68348
$69$	1.87146e19	0.921014
$70$	0	0
$71$	3.71100e19	1.35294	0.676470	−	0.736470i	$-0.263509\pi$
$71$	3.71100e19	1.35294	0.676470	+	0.736470i	$0.263509\pi$
$72$	−2.95164e19	−0.929122
$73$	−4.95703e19	−1.34999	−0.674997	−	0.737821i	$-0.735855\pi$
$73$	−4.95703e19	−1.34999	−0.674997	+	0.737821i	$0.735855\pi$
$74$	−1.34187e20	−3.16794
$75$	0	0
$76$	−6.20497e19	−1.10713
$77$	6.21229e18	0.0966272
$78$	1.27141e20	1.72700
$79$	6.84806e19	0.813735	0.406867	−	0.913487i	$-0.366621\pi$
$79$	6.84806e19	0.813735	0.406867	+	0.913487i	$0.366621\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	−8.69014e19	−0.698203
$83$	1.44041e20	1.01898	0.509490	−	0.860477i	$-0.329834\pi$
$83$	1.44041e20	1.01898	0.509490	+	0.860477i	$0.329834\pi$
$84$	3.05603e19	0.190645
$85$	0	0
$86$	−1.35260e19	−0.0659077
$87$	−5.56358e19	−0.240105
$88$	−5.30969e20	−2.03236
$89$	−2.66839e20	−0.907097	−0.453549	−	0.891232i	$-0.649842\pi$
$89$	−2.66839e20	−0.907097	−0.453549	+	0.891232i	$0.649842\pi$
$90$	0	0
$91$	−7.88065e19	−0.212142
$92$	1.65612e21	3.97484
$93$	2.93028e20	0.627824
$94$	4.49544e20	0.860854
$95$	0	0
$96$	−8.60960e20	−1.32171
$97$	−7.10430e20	−0.978178	−0.489089	−	0.872234i	$-0.662671\pi$
$97$	−7.10430e20	−0.978178	−0.489089	+	0.872234i	$0.662671\pi$
$98$	1.48490e21	1.83579
$99$	2.18703e20	0.243044
$100$	0	0
$101$	1.17701e21	1.06024	0.530120	−	0.847923i	$-0.322147\pi$
$101$	1.17701e21	1.06024	0.530120	+	0.847923i	$0.322147\pi$
$102$	1.96347e21	1.59486
$103$	−3.68602e20	−0.270251	−0.135125	−	0.990829i	$-0.543144\pi$
$103$	−3.68602e20	−0.270251	−0.135125	+	0.990829i	$0.543144\pi$
$104$	6.73565e21	4.46200
$105$	0	0
$106$	1.50697e21	0.817323
$107$	−8.75112e20	−0.430065	−0.215032	−	0.976607i	$-0.568986\pi$
$107$	−8.75112e20	−0.430065	−0.215032	+	0.976607i	$0.568986\pi$
$108$	1.07587e21	0.479525
$109$	−2.27503e21	−0.920468	−0.460234	−	0.887798i	$-0.652234\pi$
$109$	−2.27503e21	−0.920468	−0.460234	+	0.887798i	$0.652234\pi$
$110$	0	0
$111$	2.92813e21	0.978811
$112$	1.18342e21	0.360040
$113$	2.79628e21	0.774921	0.387461	−	0.921886i	$-0.373352\pi$
$113$	2.79628e21	0.774921	0.387461	+	0.921886i	$0.373352\pi$
$114$	1.89741e21	0.479359
$115$	0	0
$116$	−4.92339e21	−1.03623
$117$	−2.77438e21	−0.533598
$118$	−7.90483e21	−1.39037
$119$	−1.21703e21	−0.195911
$120$	0	0
$121$	−3.46602e21	−0.468365
$122$	−1.04385e22	−1.29378
$123$	1.89630e21	0.215726
$124$	2.59310e22	2.70951
$125$	0	0
$126$	−9.34500e20	−0.0825446
$127$	−1.79634e21	−0.146032	−0.0730161	−	0.997331i	$-0.523262\pi$
$127$	−1.79634e21	−0.146032	−0.0730161	+	0.997331i	$0.523262\pi$
$128$	−8.38130e21	−0.627490
$129$	2.95155e20	0.0203638
$130$	0	0
$131$	1.94626e22	1.14249	0.571246	−	0.820779i	$-0.306460\pi$
$131$	1.94626e22	1.14249	0.571246	+	0.820779i	$0.306460\pi$
$132$	1.93538e22	1.04891
$133$	−1.17609e21	−0.0588839
$134$	7.26748e22	3.36344
$135$	0	0
$136$	1.04020e23	4.12059
$137$	−1.04922e22	−0.384859	−0.192430	−	0.981311i	$-0.561637\pi$
$137$	−1.04922e22	−0.384859	−0.192430	+	0.981311i	$0.561637\pi$
$138$	−5.06424e22	−1.72101
$139$	−9.75832e21	−0.307411	−0.153706	−	0.988117i	$-0.549121\pi$
$139$	−9.75832e21	−0.307411	−0.153706	+	0.988117i	$0.549121\pi$
$140$	0	0
$141$	−9.80961e21	−0.265981
$142$	−1.00421e23	−2.52811
$143$	−4.99080e22	−1.16719
$144$	4.16623e22	0.905601
$145$	0	0
$146$	1.34139e23	2.52260
$147$	−3.24023e22	−0.567211
$148$	2.59120e23	4.22428
$149$	8.68462e22	1.31915	0.659577	−	0.751637i	$-0.270736\pi$
$149$	8.68462e22	1.31915	0.659577	+	0.751637i	$0.270736\pi$
$150$	0	0
$151$	−1.00167e23	−1.32271	−0.661356	−	0.750073i	$-0.730019\pi$
$151$	−1.00167e23	−1.32271	−0.661356	+	0.750073i	$0.730019\pi$
$152$	1.00521e23	1.23851
$153$	−4.28454e22	−0.492770
$154$	−1.68106e22	−0.180558
$155$	0	0
$156$	−2.45514e23	−2.30286
$157$	1.45760e23	1.27847	0.639236	−	0.769011i	$-0.279251\pi$
$157$	1.45760e23	1.27847	0.639236	+	0.769011i	$0.279251\pi$
$158$	−1.85310e23	−1.52055
$159$	−3.28841e22	−0.252532
$160$	0	0
$161$	3.13900e22	0.211407
$162$	−3.28990e22	−0.207623
$163$	−1.18785e23	−0.702737	−0.351368	−	0.936237i	$-0.614284\pi$
$163$	−1.18785e23	−0.702737	−0.351368	+	0.936237i	$0.614284\pi$
$164$	1.67810e23	0.931015
$165$	0	0
$166$	−3.89778e23	−1.90407
$167$	1.87137e23	0.858294	0.429147	−	0.903235i	$-0.358814\pi$
$167$	1.87137e23	0.858294	0.429147	+	0.903235i	$0.358814\pi$
$168$	−4.95078e22	−0.213268
$169$	3.86048e23	1.56254
$170$	0	0
$171$	−4.14040e22	−0.148110
$172$	2.61192e22	0.0878844
$173$	1.82483e23	0.577749	0.288874	−	0.957367i	$-0.406719\pi$
$173$	1.82483e23	0.577749	0.288874	+	0.957367i	$0.406719\pi$
$174$	1.50552e23	0.448662
$175$	0	0
$176$	7.49459e23	1.98091
$177$	1.72493e23	0.429589
$178$	7.22073e23	1.69501
$179$	−7.25018e23	−1.60469	−0.802347	−	0.596858i	$-0.796415\pi$
$179$	−7.25018e23	−1.60469	−0.802347	+	0.596858i	$0.796415\pi$
$180$	0	0
$181$	7.85990e23	1.54808	0.774038	−	0.633140i	$-0.218234\pi$
$181$	7.85990e23	1.54808	0.774038	+	0.633140i	$0.218234\pi$
$182$	2.13253e23	0.396410
$183$	2.27781e23	0.399743
$184$	−2.68292e24	−4.44653
$185$	0	0
$186$	−7.92941e23	−1.17315
$187$	−7.70743e23	−1.07789
$188$	−8.68085e23	−1.14790
$189$	2.03920e22	0.0255041
$190$	0	0
$191$	4.21923e23	0.472480	0.236240	−	0.971695i	$-0.424085\pi$
$191$	4.21923e23	0.472480	0.236240	+	0.971695i	$0.424085\pi$
$192$	8.50128e23	0.901201
$193$	−1.85018e24	−1.85722	−0.928608	−	0.371061i	$-0.878994\pi$
$193$	−1.85018e24	−1.85722	−0.928608	+	0.371061i	$0.878994\pi$
$194$	1.92244e24	1.82783
$195$	0	0
$196$	−2.86739e24	−2.44792
$197$	−1.91656e24	−1.55105	−0.775527	−	0.631315i	$-0.782516\pi$
$197$	−1.91656e24	−1.55105	−0.775527	+	0.631315i	$0.782516\pi$
$198$	−5.91817e23	−0.454154
$199$	−2.45101e24	−1.78397	−0.891984	−	0.452067i	$-0.850687\pi$
$199$	−2.45101e24	−1.78397	−0.891984	+	0.452067i	$0.850687\pi$
$200$	0	0
$201$	−1.58586e24	−1.03922
$202$	−3.18501e24	−1.98117
$203$	−9.33175e22	−0.0551131
$204$	−3.79153e24	−2.12666
$205$	0	0
$206$	9.97447e23	0.504991
$207$	1.10508e24	0.531748
$208$	−9.50732e24	−4.34904
$209$	−7.44813e23	−0.323975
$210$	0	0
$211$	−1.50987e24	−0.594258	−0.297129	−	0.954837i	$-0.596029\pi$
$211$	−1.50987e24	−0.594258	−0.297129	+	0.954837i	$0.596029\pi$
$212$	−2.91002e24	−1.08986
$213$	2.19131e24	0.781121
$214$	2.36808e24	0.803621
$215$	0	0
$216$	−1.74292e24	−0.536429
$217$	4.91494e23	0.144109
$218$	6.15629e24	1.71999
$219$	−2.92708e24	−0.779419
$220$	0	0
$221$	9.77732e24	2.36647
$222$	−7.92360e24	−1.82901
$223$	3.47017e24	0.764098	0.382049	−	0.924142i	$-0.375219\pi$
$223$	3.47017e24	0.764098	0.382049	+	0.924142i	$0.375219\pi$
$224$	−1.44408e24	−0.303381
$225$	0	0
$226$	−7.56682e24	−1.44802
$227$	4.24218e24	0.775028	0.387514	−	0.921864i	$-0.373334\pi$
$227$	4.24218e24	0.775028	0.387514	+	0.921864i	$0.373334\pi$
$228$	−3.66398e24	−0.639200
$229$	−5.07895e23	−0.0846255	−0.0423127	−	0.999104i	$-0.513473\pi$
$229$	−5.07895e23	−0.0846255	−0.0423127	+	0.999104i	$0.513473\pi$
$230$	0	0
$231$	3.66830e23	0.0557877
$232$	7.97591e24	1.15920
$233$	−1.02681e25	−1.42644	−0.713218	−	0.700942i	$-0.752763\pi$
$233$	−1.02681e25	−1.42644	−0.713218	+	0.700942i	$0.752763\pi$
$234$	7.50754e24	0.997083
$235$	0	0
$236$	1.52645e25	1.85399
$237$	4.04371e24	0.469810
$238$	3.29332e24	0.366080
$239$	−3.91464e24	−0.416404	−0.208202	−	0.978086i	$-0.566761\pi$
$239$	−3.91464e24	−0.416404	−0.208202	+	0.978086i	$0.566761\pi$
$240$	0	0
$241$	3.86218e23	0.0376404	0.0188202	−	0.999823i	$-0.494009\pi$
$241$	3.86218e23	0.0376404	0.0188202	+	0.999823i	$0.494009\pi$
$242$	9.37914e24	0.875189
$243$	7.17898e23	0.0641500
$244$	2.01571e25	1.72518
$245$	0	0
$246$	−5.13144e24	−0.403107
$247$	9.44839e24	0.711278
$248$	−4.20083e25	−3.03105
$249$	8.50546e24	0.588308
$250$	0	0
$251$	−7.96952e24	−0.506825	−0.253413	−	0.967358i	$-0.581553\pi$
$251$	−7.96952e24	−0.506825	−0.253413	+	0.967358i	$0.581553\pi$
$252$	1.80455e24	0.110069
$253$	1.98792e25	1.16314
$254$	4.86094e24	0.272877
$255$	0	0
$256$	−7.51267e24	−0.388396
$257$	1.73698e25	0.861981	0.430990	−	0.902357i	$-0.358164\pi$
$257$	1.73698e25	0.861981	0.430990	+	0.902357i	$0.358164\pi$
$258$	−7.98698e23	−0.0380518
$259$	4.91134e24	0.224674
$260$	0	0
$261$	−3.28524e24	−0.138625
$262$	−5.26664e25	−2.13486
$263$	−1.53148e25	−0.596453	−0.298227	−	0.954495i	$-0.596395\pi$
$263$	−1.53148e25	−0.596453	−0.298227	+	0.954495i	$0.596395\pi$
$264$	−3.13532e25	−1.17338
$265$	0	0
$266$	3.18252e24	0.110031
$267$	−1.57566e25	−0.523713
$268$	−1.40338e26	−4.48496
$269$	1.08530e25	0.333544	0.166772	−	0.985996i	$-0.446666\pi$
$269$	1.08530e25	0.333544	0.166772	+	0.985996i	$0.446666\pi$
$270$	0	0
$271$	8.52984e23	0.0242529	0.0121264	−	0.999926i	$-0.496140\pi$
$271$	8.52984e23	0.0242529	0.0121264	+	0.999926i	$0.496140\pi$
$272$	−1.46824e26	−4.01628
$273$	−4.65345e24	−0.122481
$274$	2.83923e25	0.719150
$275$	0	0
$276$	9.77922e25	2.29488
$277$	2.96382e25	0.669598	0.334799	−	0.942290i	$-0.391332\pi$
$277$	2.96382e25	0.669598	0.334799	+	0.942290i	$0.391332\pi$
$278$	2.64063e25	0.574430
$279$	1.73030e25	0.362474
$280$	0	0
$281$	2.85731e22	0.000555316 0	0.000277658	−	1.00000i	$-0.499912\pi$
$281$	2.85731e22	0.000555316 0	0.000277658		1.00000i	$0.499912\pi$
$282$	2.65451e25	0.497014
$283$	5.18615e24	0.0935594	0.0467797	−	0.998905i	$-0.485104\pi$
$283$	5.18615e24	0.0935594	0.0467797	+	0.998905i	$0.485104\pi$
$284$	1.93916e26	3.37110
$285$	0	0
$286$	1.35053e26	2.18102
$287$	3.18065e24	0.0495172
$288$	−5.08388e25	−0.763088
$289$	8.19018e25	1.18540
$290$	0	0
$291$	−4.19502e25	−0.564752
$292$	−2.59027e26	−3.36376
$293$	−5.59204e23	−0.00700584	−0.00350292	−	0.999994i	$-0.501115\pi$
$293$	−5.59204e23	−0.00700584	−0.00350292	+	0.999994i	$0.501115\pi$
$294$	8.76817e25	1.05989
$295$	0	0
$296$	−4.19775e26	−4.72556
$297$	1.29142e25	0.140322
$298$	−2.35008e26	−2.46498
$299$	−2.52179e26	−2.55365
$300$	0	0
$301$	4.95062e23	0.00467424
$302$	2.71054e26	2.47162
$303$	6.95010e25	0.612130
$304$	−1.41885e26	−1.20715
$305$	0	0
$306$	1.15941e26	0.920793
$307$	−1.16183e26	−0.891638	−0.445819	−	0.895123i	$-0.647087\pi$
$307$	−1.16183e26	−0.891638	−0.445819	+	0.895123i	$0.647087\pi$
$308$	3.24620e25	0.240764
$309$	−2.17656e25	−0.156029
$310$	0	0
$311$	−1.70436e26	−1.14177	−0.570885	−	0.821030i	$-0.693400\pi$
$311$	−1.70436e26	−1.14177	−0.570885	+	0.821030i	$0.693400\pi$
$312$	3.97733e26	2.57613
$313$	1.57708e26	0.987728	0.493864	−	0.869539i	$-0.335584\pi$
$313$	1.57708e26	0.987728	0.493864	+	0.869539i	$0.335584\pi$
$314$	−3.94430e26	−2.38896
$315$	0	0
$316$	3.57841e26	2.02757
$317$	2.57847e25	0.141332	0.0706658	−	0.997500i	$-0.477488\pi$
$317$	2.57847e25	0.141332	0.0706658	+	0.997500i	$0.477488\pi$
$318$	8.89853e25	0.471882
$319$	−5.90978e25	−0.303228
$320$	0	0
$321$	−5.16745e25	−0.248298
$322$	−8.49422e25	−0.395036
$323$	1.45914e26	0.656856
$324$	6.35291e25	0.276854
$325$	0	0
$326$	3.21436e26	1.31314
$327$	−1.34338e26	−0.531432
$328$	−2.71853e26	−1.04150
$329$	−1.64536e25	−0.0610526
$330$	0	0
$331$	−9.30107e25	−0.323846	−0.161923	−	0.986803i	$-0.551770\pi$
$331$	−9.30107e25	−0.323846	−0.161923	+	0.986803i	$0.551770\pi$
$332$	7.52676e26	2.53897
$333$	1.72903e26	0.565117
$334$	−5.06398e26	−1.60381
$335$	0	0
$336$	6.98799e25	0.207869
$337$	−4.60115e26	−1.32664	−0.663319	−	0.748337i	$-0.730853\pi$
$337$	−4.60115e26	−1.32664	−0.663319	+	0.748337i	$0.730853\pi$
$338$	−1.04466e27	−2.91977
$339$	1.65118e26	0.447401
$340$	0	0
$341$	3.11262e26	0.792876
$342$	1.12040e26	0.276758
$343$	−1.09668e26	−0.262719
$344$	−4.23133e25	−0.0983134
$345$	0	0
$346$	−4.93805e26	−1.07958
$347$	3.87832e26	0.822592	0.411296	−	0.911502i	$-0.365076\pi$
$347$	3.87832e26	0.822592	0.411296	+	0.911502i	$0.365076\pi$
$348$	−2.90721e26	−0.598267
$349$	−7.45313e25	−0.148824	−0.0744118	−	0.997228i	$-0.523708\pi$
$349$	−7.45313e25	−0.148824	−0.0744118	+	0.997228i	$0.523708\pi$
$350$	0	0
$351$	−1.63824e26	−0.308073
$352$	−9.14535e26	−1.66918
$353$	−4.90581e26	−0.869113	−0.434556	−	0.900645i	$-0.643095\pi$
$353$	−4.90581e26	−0.869113	−0.434556	+	0.900645i	$0.643095\pi$
$354$	−4.66772e26	−0.802732
$355$	0	0
$356$	−1.39435e27	−2.26020
$357$	−7.18644e25	−0.113109
$358$	1.96192e27	2.99854
$359$	−4.26796e26	−0.633474	−0.316737	−	0.948513i	$-0.602587\pi$
$359$	−4.26796e26	−0.633474	−0.316737	+	0.948513i	$0.602587\pi$
$360$	0	0
$361$	−5.73204e26	−0.802572
$362$	−2.12691e27	−2.89274
$363$	−2.04665e26	−0.270411
$364$	−4.11799e26	−0.528592
$365$	0	0
$366$	−6.16382e26	−0.746962
$367$	−1.14503e27	−1.34842	−0.674208	−	0.738542i	$-0.735515\pi$
$367$	−1.14503e27	−1.34842	−0.674208	+	0.738542i	$0.735515\pi$
$368$	3.78693e27	4.33396
$369$	1.11975e26	0.124550
$370$	0	0
$371$	−5.51563e25	−0.0579654
$372$	1.53120e27	1.56434
$373$	−1.92466e26	−0.191167	−0.0955834	−	0.995421i	$-0.530472\pi$
$373$	−1.92466e26	−0.191167	−0.0955834	+	0.995421i	$0.530472\pi$
$374$	2.08565e27	2.01414
$375$	0	0
$376$	1.40630e27	1.28412
$377$	7.49690e26	0.665729
$378$	−5.51813e25	−0.0476571
$379$	3.81406e26	0.320388	0.160194	−	0.987086i	$-0.448788\pi$
$379$	3.81406e26	0.320388	0.160194	+	0.987086i	$0.448788\pi$
$380$	0	0
$381$	−1.06072e26	−0.0843118
$382$	−1.14174e27	−0.882878
$383$	−7.69175e26	−0.578679	−0.289340	−	0.957227i	$-0.593436\pi$
$383$	−7.69175e26	−0.578679	−0.289340	+	0.957227i	$0.593436\pi$
$384$	−4.94907e26	−0.362281
$385$	0	0
$386$	5.00664e27	3.47040
$387$	1.74286e25	0.0117570
$388$	−3.71231e27	−2.43731
$389$	−6.67901e26	−0.426816	−0.213408	−	0.976963i	$-0.568456\pi$
$389$	−6.67901e26	−0.426816	−0.213408	+	0.976963i	$0.568456\pi$
$390$	0	0
$391$	−3.89447e27	−2.35826
$392$	4.64518e27	2.73841
$393$	1.14925e27	0.659618
$394$	5.18626e27	2.89830
$395$	0	0
$396$	1.14282e27	0.605590
$397$	7.11987e26	0.367428	0.183714	−	0.982980i	$-0.441188\pi$
$397$	7.11987e26	0.367428	0.183714	+	0.982980i	$0.441188\pi$
$398$	6.63249e27	3.33353
$399$	−6.94467e25	−0.0339967
$400$	0	0
$401$	−2.93203e26	−0.136192	−0.0680961	−	0.997679i	$-0.521692\pi$
$401$	−2.93203e26	−0.136192	−0.0680961	+	0.997679i	$0.521692\pi$
$402$	4.29138e27	1.94188
$403$	−3.94854e27	−1.74074
$404$	6.15037e27	2.64178
$405$	0	0
$406$	2.52520e26	0.102985
$407$	3.11034e27	1.23614
$408$	6.14230e27	2.37903
$409$	−2.98499e27	−1.12680	−0.563402	−	0.826183i	$-0.690508\pi$
$409$	−2.98499e27	−1.12680	−0.563402	+	0.826183i	$0.690508\pi$
$410$	0	0
$411$	−6.19556e26	−0.222199
$412$	−1.92611e27	−0.673379
$413$	2.89322e26	0.0986067
$414$	−2.99038e27	−0.993626
$415$	0	0
$416$	1.16014e28	3.66464
$417$	−5.76219e26	−0.177484
$418$	2.01549e27	0.605381
$419$	2.82059e26	0.0826215	0.0413108	−	0.999146i	$-0.486847\pi$
$419$	2.82059e26	0.0826215	0.0413108	+	0.999146i	$0.486847\pi$
$420$	0	0
$421$	3.73593e27	1.04097	0.520483	−	0.853872i	$-0.325752\pi$
$421$	3.73593e27	1.04097	0.520483	+	0.853872i	$0.325752\pi$
$422$	4.08576e27	1.11043
$423$	−5.79248e26	−0.153564
$424$	4.71424e27	1.21919
$425$	0	0
$426$	−5.92975e27	−1.45961
$427$	3.82056e26	0.0917560
$428$	−4.57285e27	−1.07159
$429$	−2.94702e27	−0.673878
$430$	0	0
$431$	1.76760e27	0.384923	0.192461	−	0.981305i	$-0.438353\pi$
$431$	1.76760e27	0.384923	0.192461	+	0.981305i	$0.438353\pi$
$432$	2.46012e27	0.522849
$433$	3.71253e27	0.770100	0.385050	−	0.922896i	$-0.374184\pi$
$433$	3.71253e27	0.770100	0.385050	+	0.922896i	$0.374184\pi$
$434$	−1.33000e27	−0.269283
$435$	0	0
$436$	−1.18880e28	−2.29351
$437$	−3.76345e27	−0.708812
$438$	7.92076e27	1.45643
$439$	−1.64531e27	−0.295372	−0.147686	−	0.989034i	$-0.547183\pi$
$439$	−1.64531e27	−0.295372	−0.147686	+	0.989034i	$0.547183\pi$
$440$	0	0
$441$	−1.91333e27	−0.327479
$442$	−2.64577e28	−4.42200
$443$	8.24316e27	1.34541	0.672705	−	0.739911i	$-0.265132\pi$
$443$	8.24316e27	1.34541	0.672705	+	0.739911i	$0.265132\pi$
$444$	1.53008e28	2.43889
$445$	0	0
$446$	−9.39037e27	−1.42780
$447$	5.12818e27	0.761613
$448$	1.42592e27	0.206859
$449$	−2.31679e27	−0.328323	−0.164161	−	0.986434i	$-0.552492\pi$
$449$	−2.31679e27	−0.328323	−0.164161	+	0.986434i	$0.552492\pi$
$450$	0	0
$451$	2.01430e27	0.272440
$452$	1.46118e28	1.93086
$453$	−5.91474e27	−0.763668
$454$	−1.14795e28	−1.44822
$455$	0	0
$456$	5.93566e27	0.715052
$457$	1.26565e27	0.149003	0.0745015	−	0.997221i	$-0.476263\pi$
$457$	1.26565e27	0.149003	0.0745015	+	0.997221i	$0.476263\pi$
$458$	1.37438e27	0.158132
$459$	−2.52998e27	−0.284501
$460$	0	0
$461$	−1.21042e28	−1.30040	−0.650199	−	0.759764i	$-0.725314\pi$
$461$	−1.21042e28	−1.30040	−0.650199	+	0.759764i	$0.725314\pi$
$462$	−9.92651e26	−0.104245
$463$	−9.25253e27	−0.949861	−0.474931	−	0.880023i	$-0.657527\pi$
$463$	−9.25253e27	−0.949861	−0.474931	+	0.880023i	$0.657527\pi$
$464$	−1.12580e28	−1.12985
$465$	0	0
$466$	2.77858e28	2.66545
$467$	−1.29609e28	−1.21564	−0.607822	−	0.794073i	$-0.707957\pi$
$467$	−1.29609e28	−1.21564	−0.607822	+	0.794073i	$0.707957\pi$
$468$	−1.44973e28	−1.32956
$469$	−2.65995e27	−0.238539
$470$	0	0
$471$	8.60696e27	0.738126
$472$	−2.47286e28	−2.07400
$473$	3.13522e26	0.0257173
$474$	−1.09424e28	−0.877889
$475$	0	0
$476$	−6.35952e27	−0.488147
$477$	−1.94177e27	−0.145799
$478$	1.05931e28	0.778094
$479$	−4.13344e27	−0.297022	−0.148511	−	0.988911i	$-0.547448\pi$
$479$	−4.13344e27	−0.297022	−0.148511	+	0.988911i	$0.547448\pi$
$480$	0	0
$481$	−3.94565e28	−2.71391
$482$	−1.04512e27	−0.0703350
$483$	1.85355e27	0.122056
$484$	−1.81115e28	−1.16702
$485$	0	0
$486$	−1.94265e27	−0.119871
$487$	2.58972e28	1.56386	0.781931	−	0.623365i	$-0.214235\pi$
$487$	2.58972e28	1.56386	0.781931	+	0.623365i	$0.214235\pi$
$488$	−3.26546e28	−1.92991
$489$	−7.01415e27	−0.405725
$490$	0	0
$491$	−3.59742e28	−1.99359	−0.996794	−	0.0800154i	$-0.974503\pi$
$491$	−3.59742e28	−1.99359	−0.996794	+	0.0800154i	$0.974503\pi$
$492$	9.90900e27	0.537522
$493$	1.15777e28	0.614792
$494$	−2.55676e28	−1.32910
$495$	0	0
$496$	5.92944e28	2.95431
$497$	3.67547e27	0.179296
$498$	−2.30160e28	−1.09931
$499$	−9.22436e27	−0.431400	−0.215700	−	0.976460i	$-0.569203\pi$
$499$	−9.22436e27	−0.431400	−0.215700	+	0.976460i	$0.569203\pi$
$500$	0	0
$501$	1.10502e28	0.495536
$502$	2.15658e28	0.947056
$503$	9.90722e27	0.426077	0.213039	−	0.977044i	$-0.431664\pi$
$503$	9.90722e27	0.426077	0.213039	+	0.977044i	$0.431664\pi$
$504$	−2.92338e27	−0.123130
$505$	0	0
$506$	−5.37937e28	−2.17346
$507$	2.27957e28	0.902132
$508$	−9.38665e27	−0.363866
$509$	1.26800e28	0.481486	0.240743	−	0.970589i	$-0.422609\pi$
$509$	1.26800e28	0.481486	0.240743	+	0.970589i	$0.422609\pi$
$510$	0	0
$511$	−4.90957e27	−0.178906
$512$	3.79063e28	1.35325
$513$	−2.44487e27	−0.0855111
$514$	−4.70032e28	−1.61070
$515$	0	0
$516$	1.54231e27	0.0507401
$517$	−1.04200e28	−0.335907
$518$	−1.32902e28	−0.419826
$519$	1.07755e28	0.333564
$520$	0	0
$521$	−6.03218e28	−1.79340	−0.896702	−	0.442635i	$-0.854044\pi$
$521$	−6.03218e28	−1.79340	−0.896702	+	0.442635i	$0.854044\pi$
$522$	8.88994e27	0.259035
$523$	−3.25768e28	−0.930339	−0.465169	−	0.885222i	$-0.654007\pi$
$523$	−3.25768e28	−0.930339	−0.465169	+	0.885222i	$0.654007\pi$
$524$	1.01701e29	2.84673
$525$	0	0
$526$	4.14423e28	1.11454
$527$	−6.09783e28	−1.60755
$528$	4.42548e28	1.14368
$529$	6.09757e28	1.54480
$530$	0	0
$531$	1.01856e28	0.248023
$532$	−6.14557e27	−0.146720
$533$	−2.55526e28	−0.598135
$534$	4.26377e28	0.978612
$535$	0	0
$536$	2.27348e29	5.01719
$537$	−4.28116e28	−0.926470
$538$	−2.93686e28	−0.623261
$539$	−3.44186e28	−0.716328
$540$	0	0
$541$	−2.57533e28	−0.515538	−0.257769	−	0.966207i	$-0.582987\pi$
$541$	−2.57533e28	−0.515538	−0.257769	+	0.966207i	$0.582987\pi$
$542$	−2.30820e27	−0.0453190
$543$	4.64119e28	0.893782
$544$	1.79164e29	3.38424
$545$	0	0
$546$	1.25924e28	0.228868
$547$	−3.52017e28	−0.627620	−0.313810	−	0.949486i	$-0.601606\pi$
$547$	−3.52017e28	−0.627620	−0.313810	+	0.949486i	$0.601606\pi$
$548$	−5.48265e28	−0.958948
$549$	1.34502e28	0.230792
$550$	0	0
$551$	1.11882e28	0.184785
$552$	−1.58424e29	−2.56720
$553$	6.78249e27	0.107839
$554$	−8.02018e28	−1.25121
$555$	0	0
$556$	−5.09915e28	−0.765971
$557$	−9.99130e27	−0.147280	−0.0736398	−	0.997285i	$-0.523462\pi$
$557$	−9.99130e27	−0.147280	−0.0736398	+	0.997285i	$0.523462\pi$
$558$	−4.68224e28	−0.677321
$559$	−3.97720e27	−0.0564617
$560$	0	0
$561$	−4.55116e28	−0.622317
$562$	−7.73195e25	−0.00103767
$563$	1.08345e29	1.42715	0.713575	−	0.700579i	$-0.247075\pi$
$563$	1.08345e29	1.42715	0.713575	+	0.700579i	$0.247075\pi$
$564$	−5.12595e28	−0.662741
$565$	0	0
$566$	−1.40339e28	−0.174825
$567$	1.20413e27	0.0147248
$568$	−3.14145e29	−3.77114
$569$	3.53467e28	0.416553	0.208276	−	0.978070i	$-0.433215\pi$
$569$	3.53467e28	0.416553	0.208276	+	0.978070i	$0.433215\pi$
$570$	0	0
$571$	−1.22762e29	−1.39439	−0.697194	−	0.716882i	$-0.745568\pi$
$571$	−1.22762e29	−1.39439	−0.697194	+	0.716882i	$0.745568\pi$
$572$	−2.60791e29	−2.90827
$573$	2.49141e28	0.272786
$574$	−8.60694e27	−0.0925282
$575$	0	0
$576$	5.01992e28	0.520309
$577$	5.98670e28	0.609314	0.304657	−	0.952462i	$-0.401458\pi$
$577$	5.98670e28	0.609314	0.304657	+	0.952462i	$0.401458\pi$
$578$	−2.21629e29	−2.21505
$579$	−1.09251e29	−1.07226
$580$	0	0
$581$	1.42662e28	0.135039
$582$	1.13518e29	1.05530
$583$	−3.49304e28	−0.318921
$584$	4.19624e29	3.76293
$585$	0	0
$586$	1.51322e27	0.0130912
$587$	6.04565e28	0.513740	0.256870	−	0.966446i	$-0.417309\pi$
$587$	6.04565e28	0.513740	0.256870	+	0.966446i	$0.417309\pi$
$588$	−1.69316e29	−1.41331
$589$	−5.89269e28	−0.483173
$590$	0	0
$591$	−1.13171e29	−0.895501
$592$	5.92510e29	4.60593
$593$	1.56118e28	0.119228	0.0596140	−	0.998222i	$-0.481013\pi$
$593$	1.56118e28	0.119228	0.0596140	+	0.998222i	$0.481013\pi$
$594$	−3.49462e28	−0.262206
$595$	0	0
$596$	4.53810e29	3.28691
$597$	−1.44729e29	−1.02997
$598$	6.82404e29	4.77177
$599$	4.77868e27	0.0328342	0.0164171	−	0.999865i	$-0.494774\pi$
$599$	4.77868e27	0.0328342	0.0164171	+	0.999865i	$0.494774\pi$
$600$	0	0
$601$	−6.08645e27	−0.0403815	−0.0201907	−	0.999796i	$-0.506427\pi$
$601$	−6.08645e27	−0.0403815	−0.0201907	+	0.999796i	$0.506427\pi$
$602$	−1.33965e27	−0.00873431
$603$	−9.36433e28	−0.599991
$604$	−5.23414e29	−3.29578
$605$	0	0
$606$	−1.88072e29	−1.14383
$607$	7.69225e28	0.459803	0.229902	−	0.973214i	$-0.426160\pi$
$607$	7.69225e28	0.459803	0.229902	+	0.973214i	$0.426160\pi$
$608$	1.73136e29	1.01719
$609$	−5.51031e27	−0.0318196
$610$	0	0
$611$	1.32184e29	0.737474
$612$	−2.23886e29	−1.22783
$613$	7.49794e28	0.404210	0.202105	−	0.979364i	$-0.435222\pi$
$613$	7.49794e28	0.404210	0.202105	+	0.979364i	$0.435222\pi$
$614$	3.14394e29	1.66612
$615$	0	0
$616$	−5.25885e28	−0.269335
$617$	1.41746e28	0.0713703	0.0356851	−	0.999363i	$-0.488639\pi$
$617$	1.41746e28	0.0713703	0.0356851	+	0.999363i	$0.488639\pi$
$618$	5.88982e28	0.291557
$619$	9.85030e28	0.479400	0.239700	−	0.970847i	$-0.422951\pi$
$619$	9.85030e28	0.479400	0.239700	+	0.970847i	$0.422951\pi$
$620$	0	0
$621$	6.52539e28	0.307005
$622$	4.61206e29	2.13352
$623$	−2.64284e28	−0.120212
$624$	−5.61398e29	−2.51092
$625$	0	0
$626$	−4.26762e29	−1.84567
$627$	−4.39805e28	−0.187047
$628$	7.61658e29	3.18555
$629$	−6.09336e29	−2.50626
$630$	0	0
$631$	−2.23901e29	−0.890733	−0.445367	−	0.895348i	$-0.646927\pi$
$631$	−2.23901e29	−0.890733	−0.445367	+	0.895348i	$0.646927\pi$
$632$	−5.79704e29	−2.26818
$633$	−8.91565e28	−0.343095
$634$	−6.97740e28	−0.264093
$635$	0	0
$636$	−1.71834e29	−0.629229
$637$	4.36621e29	1.57268
$638$	1.59920e29	0.566613
$639$	1.29395e29	0.450980
$640$	0	0
$641$	−2.12332e29	−0.716154	−0.358077	−	0.933692i	$-0.616567\pi$
$641$	−2.12332e29	−0.716154	−0.358077	+	0.933692i	$0.616567\pi$
$642$	1.39833e29	0.463971
$643$	−3.51724e29	−1.14812	−0.574060	−	0.818813i	$-0.694632\pi$
$643$	−3.51724e29	−1.14812	−0.574060	+	0.818813i	$0.694632\pi$
$644$	1.64026e29	0.526759
$645$	0	0
$646$	−3.94847e29	−1.22740
$647$	5.05216e29	1.54519	0.772596	−	0.634898i	$-0.218958\pi$
$647$	5.05216e29	1.54519	0.772596	+	0.634898i	$0.218958\pi$
$648$	−1.02917e29	−0.309707
$649$	1.83227e29	0.542526
$650$	0	0
$651$	2.90222e28	0.0832013
$652$	−6.20705e29	−1.75100
$653$	−5.01001e29	−1.39075	−0.695377	−	0.718645i	$-0.744763\pi$
$653$	−5.01001e29	−1.39075	−0.695377	+	0.718645i	$0.744763\pi$
$654$	3.63523e29	0.993037
$655$	0	0
$656$	3.83718e29	1.01513
$657$	−1.72841e29	−0.449998
$658$	4.45239e28	0.114083
$659$	−4.54546e29	−1.14625	−0.573127	−	0.819466i	$-0.694270\pi$
$659$	−4.54546e29	−1.14625	−0.573127	+	0.819466i	$0.694270\pi$
$660$	0	0
$661$	−4.75493e29	−1.16153	−0.580763	−	0.814073i	$-0.697246\pi$
$661$	−4.75493e29	−1.16153	−0.580763	+	0.814073i	$0.697246\pi$
$662$	2.51690e29	0.605141
$663$	5.77341e29	1.36628
$664$	−1.21934e30	−2.84027
$665$	0	0
$666$	−4.67881e29	−1.05598
$667$	−2.98614e29	−0.663421
$668$	9.77872e29	2.13860
$669$	2.04910e29	0.441152
$670$	0	0
$671$	2.41955e29	0.504834
$672$	−8.52717e28	−0.175157
$673$	1.91624e28	0.0387519	0.0193759	−	0.999812i	$-0.493832\pi$
$673$	1.91624e28	0.0387519	0.0193759	+	0.999812i	$0.493832\pi$
$674$	1.24508e30	2.47896
$675$	0	0
$676$	2.01727e30	3.89335
$677$	−6.56888e29	−1.24827	−0.624137	−	0.781315i	$-0.714549\pi$
$677$	−6.56888e29	−1.24827	−0.624137	+	0.781315i	$0.714549\pi$
$678$	−4.46813e29	−0.836016
$679$	−7.03628e28	−0.129631
$680$	0	0
$681$	2.50496e29	0.447463
$682$	−8.42284e29	−1.48157
$683$	−7.59453e29	−1.31548	−0.657739	−	0.753246i	$-0.728487\pi$
$683$	−7.59453e29	−1.31548	−0.657739	+	0.753246i	$0.728487\pi$
$684$	−2.16354e29	−0.369042
$685$	0	0
$686$	2.96765e29	0.490918
$687$	−2.99907e28	−0.0488585
$688$	5.97249e28	0.0958245
$689$	4.43112e29	0.700183
$690$	0	0
$691$	−1.51668e29	−0.232473	−0.116237	−	0.993222i	$-0.537083\pi$
$691$	−1.51668e29	−0.232473	−0.116237	+	0.993222i	$0.537083\pi$
$692$	9.53556e29	1.43957
$693$	2.16609e28	0.0322091
$694$	−1.04949e30	−1.53710
$695$	0	0
$696$	4.70970e29	0.669262
$697$	−3.94615e29	−0.552370
$698$	2.01684e29	0.278093
$699$	−6.06321e29	−0.823554
$700$	0	0
$701$	7.49249e29	0.987614	0.493807	−	0.869572i	$-0.335605\pi$
$701$	7.49249e29	0.987614	0.493807	+	0.869572i	$0.335605\pi$
$702$	4.43313e29	0.575666
$703$	−5.88837e29	−0.753293
$704$	9.03030e29	1.13812
$705$	0	0
$706$	1.32753e30	1.62403
$707$	1.16574e29	0.140507
$708$	9.01354e29	1.07040
$709$	1.32759e30	1.55338	0.776689	−	0.629884i	$-0.216897\pi$
$709$	1.32759e30	1.55338	0.776689	+	0.629884i	$0.216897\pi$
$710$	0	0
$711$	2.38777e29	0.271245
$712$	2.25885e30	2.52841
$713$	1.57277e30	1.73470
$714$	1.94467e29	0.211356
$715$	0	0
$716$	−3.78854e30	−3.99839
$717$	−2.31156e29	−0.240411
$718$	1.15492e30	1.18371
$719$	−3.41428e29	−0.344862	−0.172431	−	0.985022i	$-0.555162\pi$
$719$	−3.41428e29	−0.344862	−0.172431	+	0.985022i	$0.555162\pi$
$720$	0	0
$721$	−3.65073e28	−0.0358145
$722$	1.55111e30	1.49969
$723$	2.28058e28	0.0217317
$724$	4.10714e30	3.85731
$725$	0	0
$726$	5.53829e29	0.505291
$727$	3.77332e29	0.339323	0.169661	−	0.985502i	$-0.445733\pi$
$727$	3.77332e29	0.339323	0.169661	+	0.985502i	$0.445733\pi$
$728$	6.67116e29	0.591319
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−6.14210e28	−0.0521416
$732$	1.19026e30	0.996034
$733$	1.29548e30	1.06866	0.534329	−	0.845276i	$-0.320564\pi$
$733$	1.29548e30	1.06866	0.534329	+	0.845276i	$0.320564\pi$
$734$	3.09849e30	2.51966
$735$	0	0
$736$	−4.62104e30	−3.65193
$737$	−1.68454e30	−1.31242
$738$	−3.03007e29	−0.232734
$739$	−1.55698e30	−1.17901	−0.589504	−	0.807765i	$-0.700677\pi$
$739$	−1.55698e30	−1.17901	−0.589504	+	0.807765i	$0.700677\pi$
$740$	0	0
$741$	5.57918e29	0.410657
$742$	1.49254e29	0.108314
$743$	−9.82980e29	−0.703335	−0.351667	−	0.936125i	$-0.614385\pi$
$743$	−9.82980e29	−0.703335	−0.351667	+	0.936125i	$0.614385\pi$
$744$	−2.48055e30	−1.74997
$745$	0	0
$746$	5.20820e29	0.357215
$747$	5.02239e29	0.339660
$748$	−4.02747e30	−2.68575
$749$	−8.66734e28	−0.0569937
$750$	0	0
$751$	2.98176e30	1.90657	0.953286	−	0.302069i	$-0.0976774\pi$
$751$	2.98176e30	1.90657	0.953286	+	0.302069i	$0.0976774\pi$
$752$	−1.98499e30	−1.25161
$753$	−4.70592e29	−0.292616
$754$	−2.02868e30	−1.24398
$755$	0	0
$756$	1.06557e29	0.0635482
$757$	1.04571e30	0.615043	0.307521	−	0.951541i	$-0.400500\pi$
$757$	1.04571e30	0.615043	0.307521	+	0.951541i	$0.400500\pi$
$758$	−1.03210e30	−0.598678
$759$	1.17385e30	0.671542
$760$	0	0
$761$	−1.90796e30	−1.06177	−0.530884	−	0.847445i	$-0.678140\pi$
$761$	−1.90796e30	−1.06177	−0.530884	+	0.847445i	$0.678140\pi$
$762$	2.87034e29	0.157545
$763$	−2.25325e29	−0.121984
$764$	2.20473e30	1.17727
$765$	0	0
$766$	2.08141e30	1.08132
$767$	−2.32434e30	−1.19110
$768$	−4.43615e29	−0.224240
$769$	1.75998e30	0.877566	0.438783	−	0.898593i	$-0.355410\pi$
$769$	1.75998e30	0.877566	0.438783	+	0.898593i	$0.355410\pi$
$770$	0	0
$771$	1.02567e30	0.497665
$772$	−9.66801e30	−4.62760
$773$	3.62847e30	1.71332	0.856660	−	0.515881i	$-0.172535\pi$
$773$	3.62847e30	1.71332	0.856660	+	0.515881i	$0.172535\pi$
$774$	−4.71623e28	−0.0219692
$775$	0	0
$776$	6.01396e30	2.72654
$777$	2.90009e29	0.129715
$778$	1.80736e30	0.797551
$779$	−3.81339e29	−0.166023
$780$	0	0
$781$	2.32767e30	0.986474
$782$	1.05386e31	4.40666
$783$	−1.93990e29	−0.0800351
$784$	−6.55665e30	−2.66909
$785$	0	0
$786$	−3.10990e30	−1.23256
$787$	−3.24952e30	−1.27082	−0.635412	−	0.772174i	$-0.719170\pi$
$787$	−3.24952e30	−1.27082	−0.635412	+	0.772174i	$0.719170\pi$
$788$	−1.00149e31	−3.86473
$789$	−9.04324e29	−0.344362
$790$	0	0
$791$	2.76951e29	0.102695
$792$	−1.85137e30	−0.677454
$793$	−3.06934e30	−1.10835
$794$	−1.92666e30	−0.686577
$795$	0	0
$796$	−1.28076e31	−4.44508
$797$	−4.55638e30	−1.56066	−0.780328	−	0.625370i	$-0.784948\pi$
$797$	−4.55638e30	−1.56066	−0.780328	+	0.625370i	$0.784948\pi$
$798$	1.87925e29	0.0635263
$799$	2.04136e30	0.681048
$800$	0	0
$801$	−9.30409e29	−0.302366
$802$	7.93415e29	0.254489
$803$	−3.10922e30	−0.984325
$804$	−8.28680e30	−2.58940
$805$	0	0
$806$	1.06849e31	3.25275
$807$	6.40861e29	0.192571
$808$	−9.96363e30	−2.95528
$809$	5.77266e30	1.69012	0.845058	−	0.534675i	$-0.179566\pi$
$809$	5.77266e30	1.69012	0.845058	+	0.534675i	$0.179566\pi$
$810$	0	0
$811$	−3.28147e30	−0.936158	−0.468079	−	0.883687i	$-0.655054\pi$
$811$	−3.28147e30	−0.936158	−0.468079	+	0.883687i	$0.655054\pi$
$812$	−4.87625e29	−0.137324
$813$	5.03678e28	0.0140024
$814$	−8.41667e30	−2.30985
$815$	0	0
$816$	−8.66982e30	−2.31880
$817$	−5.93546e28	−0.0156720
$818$	8.07748e30	2.10555
$819$	−2.74781e29	−0.0707142
$820$	0	0
$821$	2.02023e30	0.506755	0.253378	−	0.967367i	$-0.418458\pi$
$821$	2.02023e30	0.506755	0.253378	+	0.967367i	$0.418458\pi$
$822$	1.67654e30	0.415202
$823$	−1.82318e30	−0.445791	−0.222896	−	0.974842i	$-0.571551\pi$
$823$	−1.82318e30	−0.445791	−0.222896	+	0.974842i	$0.571551\pi$
$824$	3.12030e30	0.753287
$825$	0	0
$826$	−7.82914e29	−0.184257
$827$	8.27709e30	1.92340	0.961701	−	0.274101i	$-0.0883802\pi$
$827$	8.27709e30	1.92340	0.961701	+	0.274101i	$0.0883802\pi$
$828$	5.77453e30	1.32495
$829$	−1.63653e30	−0.370766	−0.185383	−	0.982666i	$-0.559353\pi$
$829$	−1.63653e30	−0.370766	−0.185383	+	0.982666i	$0.559353\pi$
$830$	0	0
$831$	1.75011e30	0.386593
$832$	−1.14555e31	−2.49872
$833$	6.74285e30	1.45235
$834$	1.55927e30	0.331647
$835$	0	0
$836$	−3.89197e30	−0.807243
$837$	1.02172e30	0.209275
$838$	−7.63261e29	−0.154387
$839$	−3.42876e29	−0.0684916	−0.0342458	−	0.999413i	$-0.510903\pi$
$839$	−3.42876e29	−0.0684916	−0.0342458	+	0.999413i	$0.510903\pi$
$840$	0	0
$841$	−4.24511e30	−0.827048
$842$	−1.01095e31	−1.94516
$843$	1.68721e27	0.000320612 0
$844$	−7.88975e30	−1.48070
$845$	0	0
$846$	1.56746e30	0.286951
$847$	−3.43283e29	−0.0620693
$848$	−6.65413e30	−1.18832
$849$	3.06237e29	0.0540165
$850$	0	0
$851$	1.57162e31	2.70450
$852$	1.14506e31	1.94631
$853$	−4.51218e30	−0.757568	−0.378784	−	0.925485i	$-0.623658\pi$
$853$	−4.51218e30	−0.757568	−0.378784	+	0.925485i	$0.623658\pi$
$854$	−1.03385e30	−0.171456
$855$	0	0
$856$	7.40803e30	1.19875
$857$	6.44682e30	1.03050	0.515249	−	0.857041i	$-0.327700\pi$
$857$	6.44682e30	1.03050	0.515249	+	0.857041i	$0.327700\pi$
$858$	7.97472e30	1.25921
$859$	−2.66835e29	−0.0416212	−0.0208106	−	0.999783i	$-0.506625\pi$
$859$	−2.66835e29	−0.0416212	−0.0208106	+	0.999783i	$0.506625\pi$
$860$	0	0
$861$	1.87814e29	0.0285888
$862$	−4.78318e30	−0.719268
$863$	−5.55966e30	−0.825914	−0.412957	−	0.910750i	$-0.635504\pi$
$863$	−5.55966e30	−0.825914	−0.412957	+	0.910750i	$0.635504\pi$
$864$	−3.00198e30	−0.440569
$865$	0	0
$866$	−1.00462e31	−1.43901
$867$	4.83622e30	0.684393
$868$	2.56827e30	0.359074
$869$	4.29534e30	0.593321
$870$	0	0
$871$	2.13694e31	2.88139
$872$	1.92587e31	2.56568
$873$	−2.47712e30	−0.326059
$874$	1.01840e31	1.32449
$875$	0	0
$876$	−1.52953e31	−1.94207
$877$	−1.03383e31	−1.29704	−0.648522	−	0.761196i	$-0.724612\pi$
$877$	−1.03383e31	−1.29704	−0.648522	+	0.761196i	$0.724612\pi$
$878$	4.45225e30	0.551934
$879$	−3.30205e28	−0.00404483
$880$	0	0
$881$	−8.59158e30	−1.02760	−0.513802	−	0.857909i	$-0.671764\pi$
$881$	−8.59158e30	−1.02760	−0.513802	+	0.857909i	$0.671764\pi$
$882$	5.17751e30	0.611929
$883$	1.08791e31	1.27059	0.635297	−	0.772268i	$-0.280878\pi$
$883$	1.08791e31	1.27059	0.635297	+	0.772268i	$0.280878\pi$
$884$	5.10908e31	5.89649
$885$	0	0
$886$	−2.23062e31	−2.51404
$887$	−1.20793e31	−1.34537	−0.672687	−	0.739927i	$-0.734860\pi$
$887$	−1.20793e31	−1.34537	−0.672687	+	0.739927i	$0.734860\pi$
$888$	−2.47873e31	−2.72831
$889$	−1.77914e29	−0.0193527
$890$	0	0
$891$	7.62571e29	0.0810148
$892$	1.81331e31	1.90389
$893$	1.97268e30	0.204699
$894$	−1.38770e31	−1.42315
$895$	0	0
$896$	−8.30105e29	−0.0831570
$897$	−1.48909e31	−1.47435
$898$	6.26931e30	0.613505
$899$	−4.67560e30	−0.452232
$900$	0	0
$901$	6.84309e30	0.646610
$902$	−5.45076e30	−0.509083
$903$	2.92329e28	0.00269867
$904$	−2.36712e31	−2.15999
$905$	0	0
$906$	1.60054e31	1.42699
$907$	8.34912e30	0.735807	0.367904	−	0.929864i	$-0.380076\pi$
$907$	8.34912e30	0.735807	0.367904	+	0.929864i	$0.380076\pi$
$908$	2.21672e31	1.93112
$909$	4.10397e30	0.353413
$910$	0	0
$911$	−1.08094e30	−0.0909621	−0.0454811	−	0.998965i	$-0.514482\pi$
$911$	−1.08094e30	−0.0909621	−0.0454811	+	0.998965i	$0.514482\pi$
$912$	−8.37814e30	−0.696950
$913$	9.03473e30	0.742971
$914$	−3.42490e30	−0.278428
$915$	0	0
$916$	−2.65397e30	−0.210860
$917$	1.92763e30	0.151407
$918$	6.84620e30	0.531620
$919$	−9.75424e30	−0.748825	−0.374413	−	0.927262i	$-0.622156\pi$
$919$	−9.75424e30	−0.748825	−0.374413	+	0.927262i	$0.622156\pi$
$920$	0	0
$921$	−6.86047e30	−0.514787
$922$	3.27543e31	2.42993
$923$	−2.95278e31	−2.16578
$924$	1.91685e30	0.139005
$925$	0	0
$926$	2.50376e31	1.77492
$927$	−1.28523e30	−0.0900835
$928$	1.37376e31	0.952047
$929$	−2.09651e31	−1.43659	−0.718294	−	0.695739i	$-0.755077\pi$
$929$	−2.09651e31	−1.43659	−0.718294	+	0.695739i	$0.755077\pi$
$930$	0	0
$931$	6.51600e30	0.436525
$932$	−5.36553e31	−3.55423
$933$	−1.00641e31	−0.659201
$934$	3.50725e31	2.27156
$935$	0	0
$936$	2.34857e31	1.48733
$937$	−2.80685e31	−1.75774	−0.878868	−	0.477065i	$-0.841701\pi$
$937$	−2.80685e31	−1.75774	−0.878868	+	0.477065i	$0.841701\pi$
$938$	7.19790e30	0.445734
$939$	9.31249e30	0.570265
$940$	0	0
$941$	4.05224e29	0.0242664	0.0121332	−	0.999926i	$-0.496138\pi$
$941$	4.05224e29	0.0242664	0.0121332	+	0.999926i	$0.496138\pi$
$942$	−2.32907e31	−1.37927
$943$	1.01780e31	0.596061
$944$	3.49042e31	2.02149
$945$	0	0
$946$	−8.48398e29	−0.0480555
$947$	−4.53524e30	−0.254054	−0.127027	−	0.991899i	$-0.540543\pi$
$947$	−4.53524e30	−0.254054	−0.127027	+	0.991899i	$0.540543\pi$
$948$	2.11302e31	1.17062
$949$	3.94423e31	2.16106
$950$	0	0
$951$	1.52256e30	0.0815978
$952$	1.03024e31	0.546075
$953$	−1.19767e31	−0.627857	−0.313929	−	0.949447i	$-0.601645\pi$
$953$	−1.19767e31	−0.627857	−0.313929	+	0.949447i	$0.601645\pi$
$954$	5.25449e30	0.272441
$955$	0	0
$956$	−2.04557e31	−1.03755
$957$	−3.48967e30	−0.175069
$958$	1.11852e31	0.555017
$959$	−1.03918e30	−0.0510029
$960$	0	0
$961$	3.80040e30	0.182488
$962$	1.06770e32	5.07122
$963$	−3.05133e30	−0.143355
$964$	2.01816e30	0.0937879
$965$	0	0
$966$	−5.01575e30	−0.228074
$967$	6.74111e30	0.303217	0.151608	−	0.988441i	$-0.451555\pi$
$967$	6.74111e30	0.303217	0.151608	+	0.988441i	$0.451555\pi$
$968$	2.93406e31	1.30550
$969$	8.61607e30	0.379236
$970$	0	0
$971$	−2.97823e31	−1.28279	−0.641397	−	0.767210i	$-0.721645\pi$
$971$	−2.97823e31	−1.28279	−0.641397	+	0.767210i	$0.721645\pi$
$972$	3.75133e30	0.159842
$973$	−9.66489e29	−0.0407392
$974$	−7.00785e31	−2.92224
$975$	0	0
$976$	4.60917e31	1.88105
$977$	−2.35144e31	−0.949380	−0.474690	−	0.880153i	$-0.657440\pi$
$977$	−2.35144e31	−0.949380	−0.474690	+	0.880153i	$0.657440\pi$
$978$	1.89805e31	0.758140
$979$	−1.67371e31	−0.661395
$980$	0	0
$981$	−7.93254e30	−0.306823
$982$	9.73472e31	3.72523
$983$	−1.94366e31	−0.735882	−0.367941	−	0.929849i	$-0.619937\pi$
$983$	−1.94366e31	−0.735882	−0.367941	+	0.929849i	$0.619937\pi$
$984$	−1.60526e31	−0.601309
$985$	0	0
$986$	−3.13295e31	−1.14880
$987$	−9.71569e29	−0.0352487
$988$	4.93720e31	1.77228
$989$	1.58419e30	0.0562659
$990$	0	0
$991$	−5.08101e31	−1.76676	−0.883378	−	0.468662i	$-0.844736\pi$
$991$	−5.08101e31	−1.76676	−0.883378	+	0.468662i	$0.844736\pi$
$992$	−7.23547e31	−2.48940
$993$	−5.49219e30	−0.186973
$994$	−9.94594e30	−0.335034
$995$	0	0
$996$	4.44448e31	1.46588
$997$	−2.17692e31	−0.710465	−0.355233	−	0.934778i	$-0.615598\pi$
$997$	−2.17692e31	−0.710465	−0.355233	+	0.934778i	$0.615598\pi$
$998$	2.49614e31	0.806117
$999$	1.02098e31	0.326270

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.m.1.1		10
5.2	odd	4		15.22.b.a.4.1	✓	20
5.3	odd	4		15.22.b.a.4.20	yes	20
5.4	even	2		75.22.a.n.1.10		10
15.2	even	4		45.22.b.d.19.20		20
15.8	even	4		45.22.b.d.19.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.b.a.4.1	✓	20	5.2	odd	4
15.22.b.a.4.20	yes	20	5.3	odd	4
45.22.b.d.19.1		20	15.8	even	4
45.22.b.d.19.20		20	15.2	even	4
75.22.a.m.1.1		10	1.1	even	1	trivial
75.22.a.n.1.10		10	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q-2706.03 q^{2} +59049.0 q^{3} +5.22544e6 q^{4} -1.59788e8 q^{6} +9.90426e7 q^{7} -8.46523e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q-2706.03 q^{2} +59049.0 q^{3} +5.22544e6 q^{4} -1.59788e8 q^{6} +9.90426e7 q^{7} -8.46523e9 q^{8} +3.48678e9 q^{9} +6.27235e10 q^{11} +3.08557e11 q^{12} -7.95684e11 q^{13} -2.68012e11 q^{14} +1.19486e13 q^{16} -1.22879e13 q^{17} -9.43534e12 q^{18} -1.18746e13 q^{19} +5.84836e12 q^{21} -1.69731e14 q^{22} +3.16934e14 q^{23} -4.99864e14 q^{24} +2.15314e15 q^{26} +2.05891e14 q^{27} +5.17541e14 q^{28} -9.42196e14 q^{29} +4.96245e15 q^{31} -1.45804e16 q^{32} +3.70376e15 q^{33} +3.32515e16 q^{34} +1.82200e16 q^{36} +4.95881e16 q^{37} +3.21329e16 q^{38} -4.69843e16 q^{39} +3.21140e16 q^{41} -1.58258e16 q^{42} +4.99847e15 q^{43} +3.27758e17 q^{44} -8.57633e17 q^{46} -1.66127e17 q^{47} +7.05554e17 q^{48} -5.48736e17 q^{49} -7.25591e17 q^{51} -4.15780e18 q^{52} -5.56895e17 q^{53} -5.57147e17 q^{54} -8.38418e17 q^{56} -7.01180e17 q^{57} +2.54961e18 q^{58} +2.92119e18 q^{59} +3.85749e18 q^{61} -1.34285e19 q^{62} +3.45340e17 q^{63} +1.43970e19 q^{64} -1.00225e19 q^{66} -2.68566e19 q^{67} -6.42099e19 q^{68} +1.87146e19 q^{69} +3.71100e19 q^{71} -2.95164e19 q^{72} -4.95703e19 q^{73} -1.34187e20 q^{74} -6.20497e19 q^{76} +6.21229e18 q^{77} +1.27141e20 q^{78} +6.84806e19 q^{79} +1.21577e19 q^{81} -8.69014e19 q^{82} +1.44041e20 q^{83} +3.05603e19 q^{84} -1.35260e19 q^{86} -5.56358e19 q^{87} -5.30969e20 q^{88} -2.66839e20 q^{89} -7.88065e19 q^{91} +1.65612e21 q^{92} +2.93028e20 q^{93} +4.49544e20 q^{94} -8.60960e20 q^{96} -7.10430e20 q^{97} +1.48490e21 q^{98} +2.18703e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 645 q^{2} + 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} - 115357290 q^{7} - 314142735 q^{8} + 34867844010 q^{9}+O(q^{10}) \) 10 * q - 645 * q^2 + 590490 * q^3 + 10043205 * q^4 - 38086605 * q^6 - 115357290 * q^7 - 314142735 * q^8 + 34867844010 * q^9	\( 10 q - 645 q^{2} + 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} - 115357290 q^{7} - 314142735 q^{8} + 34867844010 q^{9} + 5976221790 q^{11} + 593041212045 q^{12} - 842570747430 q^{13} + 1605059874570 q^{14} + 6061861547825 q^{16} - 12910340404230 q^{17} - 2248975938645 q^{18} - 76155422176280 q^{19} - 6811732617210 q^{21} - 461780887241010 q^{22} - 82640920915920 q^{23} - 18549814359015 q^{24} + 286555331159670 q^{26} + 20\!\cdots\!90 q^{27}+ \cdots + 20\!\cdots\!90 q^{99}+O(q^{100}) \) 10 * q - 645 * q^2 + 590490 * q^3 + 10043205 * q^4 - 38086605 * q^6 - 115357290 * q^7 - 314142735 * q^8 + 34867844010 * q^9 + 5976221790 * q^11 + 593041212045 * q^12 - 842570747430 * q^13 + 1605059874570 * q^14 + 6061861547825 * q^16 - 12910340404230 * q^17 - 2248975938645 * q^18 - 76155422176280 * q^19 - 6811732617210 * q^21 - 461780887241010 * q^22 - 82640920915920 * q^23 - 18549814359015 * q^24 + 286555331159670 * q^26 + 2058911320946490 * q^27 - 1520428218531930 * q^28 - 2553653160731010 * q^29 - 5579671364724940 * q^31 - 29255856594217815 * q^32 + 352889920477710 * q^33 + 36285968426431010 * q^34 + 35018490530045205 * q^36 - 1797908431222890 * q^37 - 10684960744876320 * q^38 - 49752960064994070 * q^39 - 157319729506021440 * q^41 + 94777180533483930 * q^42 + 148924853700743400 * q^43 + 49118820773767290 * q^44 - 238453286038259540 * q^46 + 43171138083158100 * q^47 + 357946862537518425 * q^48 + 453692886452454950 * q^49 - 762342690529377270 * q^51 - 5077473324432504030 * q^52 - 1835165786666252970 * q^53 - 132799780201048605 * q^54 - 5146515624013530450 * q^56 - 4496901524087157720 * q^57 - 3684042125074285440 * q^58 + 4936135121536323630 * q^59 - 4443080871007408300 * q^61 - 39366851097921260280 * q^62 - 402225999313633290 * q^63 - 14877283608188093455 * q^64 - 27267699610694399490 * q^66 - 33664559574358232580 * q^67 - 44791395195934111290 * q^68 - 4879863739164160080 * q^69 - 74662724971386630180 * q^71 - 1095347988085476735 * q^72 - 16062862692682477620 * q^73 - 130249023546610518030 * q^74 - 258736483662380220720 * q^76 - 56216747719240740180 * q^77 + 16920805749647353830 * q^78 - 49130708010799643300 * q^79 + 121576654590569288010 * q^81 - 117185080045656261930 * q^82 - 118408383608453695560 * q^83 - 89779765876091934570 * q^84 - 1007271272794707382260 * q^86 - 150790665488005409490 * q^87 - 41803519849182967710 * q^88 - 572103976713991581780 * q^89 - 1669810835257196575140 * q^91 + 1206545757146297440260 * q^92 - 329474014415642982060 * q^93 - 1671319720124696594920 * q^94 - 1727529076031967757935 * q^96 + 1299275526640742173920 * q^97 + 2683959884079093505455 * q^98 + 20837796914288297790 * q^99

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